粘弹性大变形动力响应的有限元分析

本文基于Ｔｏｔａｌ Ｌａｇｒａｎｇｉａｎ增量叠加方法，采用Ｋｉｒｃｈｈｏｆｆ应力增量和Ｇｒｅｅｎ应变增量表示的动力虚功方程和Ｋｉｒｃｈｈｏｆｆ应力－Ｇｒｅｅｎ应变的单积分型本构关系，导出粘弹性大变形的动力变分方程。依此采用Ｎｅｗｍａｒｋ法和八节点轴对称等参数元与二十节点三维等参数元编制了轴对称及三维问题的动力响应计算程序，典型例题的计算结果表明分析符合结构的物理性质。     

粘弹性大变形动力响应的有限元分析

本文基于ＴｏｔａｌＬａｇｒａｎｇｉａｎ增量叠加方法，采用Ｋｉｒｃｈｈｏｆｆ应力增量和Ｇｒｅｅｎ应变增量表示的动力虚功方程和Ｋｉｒｃｈｈｏｆｆ应力－Ｇｒｅｅｎ应变的单积分型本构关系，导出粘弹性大变形的动力变分方程。依此采用Ｎｅｗｍａｒｋ法和八节点轴对称等参数元与二十节点三维等参数元编制了轴对称及三维问题的动力响应计算程序，典型例题的计算结果表明分析符合结构的物理性质。     

关于几种新的极分解计算方法

对变形梯度极分解的计算方法进行了分析,给出极分解计算的四种新方法：（１）增量叠加法;（２）基于伸长张量不变量（近似）计算法;（３）确定主转动轴计算法;（４）坐标变换法．增量叠加极分解计算方法将为建立以伸长张量为应变度量的大变形大转动有限元分析方法提供基础．本文还给出了伸长张量物质时间导数的简洁表达式     

基于拟应变法的壳单元大变形分析及在冲击动力问题中的应用

为了简便有效地解决板壳结构的大变形问题，本文针对八节点相对自由度壳单元进行研究。该单元的位移场由壳的中面节点位移和上表面节点的相对位移组成，不带有转动变量。所有的研究都是基于完全的三维位移、应力、应变场。采用拟应变法，对应变场另行假设，能够改善该单元在大变形情况下的计算精度。通过引入Wilson非协调模式，构造了大变形情况下的拟应变场表达式，给出了该单元用于解决非线性动力分析问题的有限元求解方程。通过算例表明，本文针对相对自由度壳单元提出的方法及推导的公式，能够解决冲击动力问题中的大变形问题。     

极分解的一类近似计算与比较

基于级数展开给出了极分解中右伸长张量 的级数表示,通过对级数的项的选取得到右伸长张量的不同近似表达式。针对不同级数展开表示,得到表达式最小误差的级数展开形式。进而结合一些简单实例,验证误差了近似公式的有效性。最后与黄模佳等关于计算右伸长张量 和转动张量 的近似表达式进行了比较,本文的级数展开方式得到的右伸长张量 和转动张量 的近似表达式不但简洁,而且计算精度更高、适用范围更广。     

关于几种新的析分解的计算方法

对变形樟度极分解的计算方法进行了分析，给出了极分解计算的四种新方法：（１）增量叠加法；（２）基于伸长张量不变量（近似）计算法；（３）确定主转动轴计算法；（４）坐标变换法。增加叠加极分解计算方法将为建立以伸长张量为变应度量的大变形大转动有限分析方法提供基础。本文还给出了伸长张量物质时间导数的简洁表达式。     

结构大变形分析中非保守载荷虚功的计算

已有的非保守大变形虚位移原理表达式在形式上是完美的,但其非保守载荷的虚功项并不能直接用于有限元数值计算。本文对此项的计算作了进一步的推导,给出了能用于实际计算的公式。对常见的非保守法向载荷,公式十分简单。对小应变大转动变形,公式也有非常简单的形式。     

各向异性介质中的Green函数解

本文首先指出Ｗａｎｇ与Ａｃｈｅｎｂａｃｈ和Ｈａｎｙｇａ关于任意各向异性，不均匀（但性质渐变）的线弹性介质中的Ｇｒｅｅｎ函数解并非完备解，而是高频条件下的近似解，并以较简洁的步骤及三维Ｒａｄｏｎ变换，得到比文献（１），（２）更合理的Ｇｒｅｅｎ函数在高频近似下显示解。在此基础上具体讨论物均匀，横观各向同性介质中的Ｇｒｅｅｎ函数完备解。     

海底控制点水声定位测距误差的近似数学表达式

针对水声定位声线跟踪法计算换能器到应答器斜距值测距误差规律难以把握的问题，研究了测距误差的近似数学表达式。首先分析了控制点定位声速不确定性引起的测距误差机理；其次基于声速剖面和传播时间的线性理论（ST定理）推导了测距误差的近似数学表达式，并分析了使用harmonic平均声速代替有效声速对测距误差数学表达式结果的影响。仿真试验结果表明，该数学表达式对测距误差估值的绝对误差随斜距值（或初始入射角）的增大而增大，在水深为3000 m，初始入射角在80°以内时，估值的绝对误差在1 cm以内，当初始入射角达到86°左右时，估值的绝对误差约7 cm，为水下静态目标定位以及水下载体动态导航定位中的声速误差修正问题提供了新的解决思路。     

半无限弹性空间域内点加振格林函数的计算

本文给出了满足全部自由面边界条件的半无限弹性空间域内点加振的Ｇｒｅｅｎ函数，利用变形的Ｈａｎｋｅｌ函数，在复数域内进行无限积分的有限化，从而使Ｇｒｅｅｎ函数的计算变得比较简单和方便。     

An approach to approximate analysis of deformation and strain

An approach to approximate analysis of deformation and strain is developed from recent work on Cauchy mean rotation [1,2]. The approximate expressions of Green strain usually employed in solving problems of nonlinear mechanics and their errors are discussed. The estimation of error is strictly based on the definition of small strain and medium or large rotation.     

Validation of the tangent formulation for the solution of the non-linear Eshelby inclusion problem

An extension of the Eshelby problem for non-linear viscous materials is considered. An ellipsoidal heterogeneity is embedded in an infinite matrix. The material properties are assumed to be uniform within the ellipsoid and in the matrix. The problem of determining the average strain rate in the ellipsoid in terms of the overall applied strain rate is solved in an approximate way. The method is based on the non-incremental tangent formulation of the non-linear matrix behavior [Acta Metall. 35 (1987) 2983]. In the present work this approximate solution is verified with a good agreement by comparing to finite element calculations for various inclusion shapes and loading conditions.     

Boundary Layer Solutions in Elastic Solids

Circumferential shear deformation in an annular domain is studied for a large class of incompressible isotropic elastic materials. It is demonstrated that large strains are confined in a region adjacent to a boundary, in analogy to the boundary layer phenomenon in fluid mechanics. The size of this region is quantified. An approximate solution technique for the deformation of nonlinear elastic solids, proposed by Rajagopal [7], is further studied. In this solution, akin to the boundary layer approximation in classical fluid mechanics, the full nonlinear problem is solved in a relatively small region of large strain, while the linearized problem is solved in the remaining region. Error estimates for the approximate solution are obtained.     

Construction of approximate solutions of boundary value impact strain problems

The method for constructing approximate solutions of boundary value problems of impact strain dynamics in the form of ray expansions behind the strain discontinuity fronts is generalized to the case of curvilinear and diverging rays. This proposed generalization is illustrated by an example of dynamics of an antiplane motion of an elastic medium. The ray method is one of the methods for constructing approximate solutions of nonstationary boundary value problems of strain dynamics. It was proposed in [1, 2] and then widely used in nonstationary problems of mathematical physics involving surfaces on which the desired function or its derivatives have discontinuities [3–7]. A complete, qualified survey of papers in this direction can be found in [8]. This method is based on the expansion of the solution in a Taylor-type series behind the moving discontinuity surface rather than in a neighborhood of a stationary point. The coefficients of this series are the jumps of the derivatives of the unknown functions, for which, as a consequence of the compatibility conditions, one can obtain ordinary differential equations, i.e., discontinuity damping equations. In the case where the problem with velocity discontinuity surfaces is considered in a nonlinear medium, this method cannot be used directly, because one cannot obtain the damping equation. A modification of this method for the purpose of using it to solve problems of that type was proposed in [9–11], where, as an example, the solutions of several one-dimensional problems were considered. In the present paper, we show how this method can be transferred to the case of multidimensional impact strain problems in which the geometry of the ray is not known in advance and the rays become curvilinear and diverging. By way of example, we consider a simple problem on the antiplane motion of a nonlinearly elastic incompressible medium.     

On the asymptotics of the stress and strain fields near a crack tip

The problem is solved under the plane strain conditions for a crack of general form, which in general is neither a mode I nor a mode II crack. We assume that the strains are small and the material is nonlinearly elastic. The mathematical statement of the problem is reduced to the eigenvalue problem for a system of ordinary nonlinear differential equations. Its solution is obtained numerically. We show that, for an incompressible material with power-law relations between the stress and strain deviators, the solution (the well-known HRR-asymptotics [1, 2]) exists only for mode I and II cracks. In the general case, we can only speak of approximate solutions. A similar conclusion can be made for different-modulus materials. We analyze the results of the preceding papers [1–7], where specific cases of the problem were considered.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

Crack tip asymptotic character of anti-plane stress and strain rate for linear fractional constitutive relations

Linear-fractional strain rate and stress relations are used to simulate materials undergoing steady state creep. The crack tip asymptotic character of the stress and strain rate field is obtained in exact and approximate form. In the limit as the radial distance emanating from the crack tip approaches zero, the stress field corresponds to that for an ideal plastic material while the exact and approximate solutions tend to coincide. Discussed is the nonhomogeneous singular character of the strain rate field that possess different orders of singularities in a circular region around the crack tip.     

Generalized stress/strain rate relations for creep of thin shells; a comparison of “exact” and approximate relations

A set of approximate generalized stress/strain rate relations which has been used for the stationary creep analysis of thin shells is compared with the corresponding ‘exact’ relations. The comparison is made by computing the functions from which the relations are derived and plotting the corresponding surfaces. Results are included for a limiting condition in which the stress/strain rate relations become those for a rigid-plastic material obeying the von Mises yield condition and associated flow rule. Although the comparison is made only for conditions valid in a cylindrical shell under rotationally symmetric loading, it indicates the errors which are likely to occur when the approximate relations are used in stationary creep analysis.     

Concentration of stresses and strains in a notched cyclinder of a viscoplastic material under harmonic loading

Certain aspects of the correct definitions of stress and strain concentration factors for elastic-viscoplastic solids under cyclic loading are discussed. Problems concerning the harmonic kinematic excitation of cylindrical specimens with a lateral V-notch are examined. The behavior of the material of a cylinder is modeled using generalized flow theory. An approximate model based on the concept of complex moduli is used for comparison. Invariant characteristics such as stress and strain intensities and maximum principal stress and strain are chosen as constitutive quantities for concentration-factor definitions. The behavior of time-varying factors is investigated. Concentration factors calculated in terms of the amplitudes of the constitutive quantities are used as representative characteristics over the cycle of vibration. The dependences of the concentration factors on the loads are also studied. The accuracy of Nueber's and Birger's formulas is evaluated. The solution of the problem in the approximate formulation agrees with its solution in the exact formulation. The possibilities of the approximate model for estimating low-cycle fatigue are evaluated. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 15–22, February, 1999.     

Accurate and approximate integrations of Drucker–Prager plasticity with linear isotropic and kinematic hardening

Up to now, some explicit approximate integration schemes based on exponential maps, for non-hardening material obeying Drucker–Prager’s criterion, have been presented. Two new exponential-based approximate formulations, for associative Drucker–Prager plasticity are developed in this article. Both are consistent and explicit algorithms. The linear isotropic and Prager’s kinematic hardening behavior are assumed. Furthermore, an accurate solution for the constitutive equations is derived. The accuracies of the suggested approximate algorithms are assessed by creating related iso-error maps. In addition, by using piecewise strain load histories, and calculating computation times, the robustness and efficiency of the formulations are demonstrated.